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Some Satirical Mathematics

The STEM crowd (I’ll call them Stemites) is very Cathedral-minded in that each contributor is envisioned as adding to a giant edifice, like a cathedral or tower, the construction of which spans maybe hundreds of years, if not thousands.

Each individual submits but a chip to the old block which gradually assembles before our eyes, perhaps according to God’s plan or at least in accordance with inviolable laws.

Aside: the idea behind much of science was we could finally get rid of The Enforcer and just keep the laws themselves, the law of gravity for example, as self-enforcing. “Rules without a Ruler” has been the battle cry, of invisible hand cybernetics versus a more manual model of monarchy. Orderly society but without top-down tyranny has been the rallying cry of our novus ordo seclorum.

In the Humanities we’re trained somewhat differently. Instead of Cathedralists, we’re more individual traders, merchants of ideas. We frequent a bazaar of perpetual renegotiation, amidst a post Babel Tower confusion of tongues. Instead of humans as extras, obedient to the one Director with the top-down blueprints, we have individual human beings as stars of our show.

Any given Thinker is allowed to have their fifteen minutes of fame (some get more), to take the floor (the stage) and soliloquize in a Private Language, unique to that one individual. Meanwhile the highly schooled members of the audience take delight in trying to decode and perhaps adapt said Thinking to their own private language needs.

Instead of sharing a gene pool (which we also do) we’re sharing a meme pool (or call it civilization, or social media).

In some cases, said thinker’s Thinking proves “seminal” meaning the memes spread everywhere and prove to have self-propagating powers.

In memetics we might speak of memeplexes or complexes “going viral”. For example, the familiar-to-many idea of “a superpower USA” is such a complex, oft named “the military-industrial” complex.

Another example: Heidegger, the Nazi, has provided some seminal thinking. Folks have picked over his Private Language like crazy for decades, cannibalizing quite a lot of it for private reuse. I don’t see this practice regarding Heidegger’s thinking ending any time soon, even with Peter Sloterdijk somewhat eclipsing Heidegger’s reputation as the quintessential German language philosopher.

Instead of the Cathedral Model (per STEM), I’ll call this humanities model the Glass Bead Game Model, citing Hermann Hesse. The humanities model features partially overlapping private languages (namespaces, scenario adventures), with some of them tapped as a cornucopia of new ideas, such as Marxism, in turn indebted to Hegelianism.

The Buckminster Fuller scenario ventures into the heart of STEM territory with the mindset of a Glass Bead Game player. Bucky didn’t envision himself as adding a chip to the Cathedral so much as starting over from scratch, determined to redesign his own language from the ground up, given the one he’d inherited had led him to the brink of suicide. He started his project by clamming up and doing minimal speaking. He would reorient himself by choosing his words differently.

By the time Fuller wrote about “4D” in Synergetics he wasn’t bending over backwards to fit his meaning to any prevailing orthodoxy. Quite the contrary in fact. He was pioneering a new Private Language, one other players might glean from. He was quite aware of Einstein’s contributions and thought a lot about time. However he was dedicated to doing his own thinking and his meaning of “4D” evolved independently.

In the Humanities, coming up with a Private Language (PL) is not even that quirky to attempt, it’s more or less expected. However perhaps since the demise of Natural Philosophy as a discipline, most in the Humanities had shied away from using much core STEM terminology, such as “gravity” and “radiation” (or frequency, entropy, dimension, angle, energy, information, experience, physical — all index key words in Synergetics).

Usually repurposing goes the other way, as in the case of quarks (for example). The STEM folks ransack the shared vocabulary for words to repurpose, but to go the other direction is considered “popularization” at best. The anchoring meanings are believed to be literal truths, not metaphorical verities, in the view of most Stemites.

Fuller diagnoses our core cultural communication problem in Operating Manual, proclaiming the Great Pirates have all died, leaving no one at the helm so to speak, on a ship full of deliberately overspecialized not-my-department types.

The glue languages had become weak, setting our mostly prose-only readers adrift in a disintegrating space of obsolete coping mechanisms.

The C.P. Snow chasm between the two cultures had grown wide. Synergetics communicates a mostly-prose result of Bucky’s lifelong project, to develop a stronger glue language based in a deliberately remote vocabulary, in hopes of thereby bridging said chasm.

Given language is public property, Bucky was happy to fish in the dictionary and repurpose any word he felt he could aesthetically (ethically) incorporate within his namespace, his Private Language, branded in concert with a host of 4D artifacts (4D car, 4D house…).

He openly discusses his “deliberately remote vocabulary” strategy and then notes how it shows some “merging traffic pattern” tendencies vis-a-vis what the Cathedralists had been up to in the meantime i.e. it might also provide some chips after all, and not just in architecture.

Time would tell more. We have more hindsight today in 2020, when Education Automation seems more relevant than ever.

Many Stemites are still convinced that their proposition “space is three dimensional” is a self-evident reality, which Euclid would have included among his axioms if only he had formalized his idea of dimension more as we do today. They see nothing “ethnic” about declaring space to be “3D” and treat this contention as one of the pillars of their great cathedral.

The three prime, elementary, positive, normally shown as rightwardly or upwardly pointing, so-called “basis vectors” define three mutually perpendicular edges of the of a right tetrahedron, with a common vertex at the origin. We’re taught the thumb and two finger rule, using our left or right hand.

Then, thanks to scalar multiplication (ordinary numbers times vectors), these primary pointers are able to shrink and grow.

Thanks to vector addition, the resulting longer-shorter vectors may then be assembled, tip-to-tail, to point anywhere within that first octant. We’ve spanned all points in the octant (+, +, +) meaning so far our scalars have all been “positive” like the vectors themselves.

Eight such right tetrahedrons, all sharing a corner at the origin (0,0,0) make the XYZ octahedron of six directional thinking. We usually think of a hexahedron, the octahedron’s six-sided dual and we do not simply “close the lid” once we have these three vectors, to define a four faceted shape.

Any four non-coplanar points define a volume with six edges, yet conceptual space (the space of containers) is not thereby “4D” unless you’re in a different namespace, such as the namespace of Synergetics.

So what about the other seven octants? We may label all eight as follows:
(+, +, +) (+, +, -) (+, -, +)(-, +, +)
(+, -, -)(-, +, -)(-, -, +) and (-, -, -).

The XYZ apparatus divvies space into these eight regions and needs a mix of positive and negative vectors to surround the origin with a growing or shrinking sphere of dots (points) which are also 4D in Synergetics i.e. they occupy volume, although perhaps in a higher frequency than we’re currently tuning in.

Thanks to -1 being considered just another scalar, despite the lurking negation operator (-), we gain the power not only to resize a vector (stretch and shrink it), but to reorient it as well, reversing its direction.

Thanks to the negation operator, we’re able to rotate any vector 180 degrees in its plane. Rightward becomes leftward whereas upward becomes downward. Vectors can flip!

By means of negating (rotating, flipping), a second set of vectors, called negative, not-elementary, not-prime vectors, get to shoulder equal work for unequal pay (lesser status). One or more of the not-primes is required, to address any point outside of the first octant i.e. in 7/8ths of XYZ space.

The oppositely pointing vectors are indistinguishable from their positive counterparts, but for a certain genesis story, which insists they came later and don’t themselves constitute additional independent directions in any way.

Space is three dimensional because only three of the vectors get to count as primary. The other three get to play second fiddle, as mere inverses.

Analyzing this orthodoxy from a more psychoanalytical point of view, we note the broken symmetry of a first class versus a second class, owners versus slaves.

The XYZ grid is inherently classist, reflecting its origins in feudalist thinking.

The elementary unit vectors are the noble courtiers, the blue blooded royals, whereas the helper vectors are but serfs. The negative team vectors are mere pawns, next to the privileged positive team vectors.

In Fuller’s 4D language game, instead of six spokes, half of them “negative”, we have only four spokes to the corners of a tetrahedron. The arrangement is more economical (fewer spokes), and the status of these four vectors is not two tier.

They’re all basis vectors and, combined with scalar multiplication sans the negation operator, reach all four quadrants of space via vector addition. The negation operator is still present, but is not invoked out of necessity. Four basis vectors without their negatives do the work of three vectors with their negatives. That’s four versus six vectors.

Our length modulating scalars don’t need the extra power or “trick” of inducing rotation, thereby sneakily introducing the XYZ servant vectors that compensate for the primary vectors’ inadequacy.

One could say the positives suffer from erectile dysfunction in XYZ, in being able to service only one of the eight neighborhoods (the exclusive “Purely Positive” neighborhood).

Our positive and zero scalars grow or shrink the basis vectors exactly along the direction they’re already pointing. That operation, plus vector addition, gives us omni-symmetrical access to the space around the origin at (0,0,0,0).

Ergo we say space is primitively four directional or dimensional (4D), and not six directional pretending to be just three dimensional (3D).

Note my PL is not exactly mirroring how Fuller does his polemics in Synergetics, but he’s likewise committed to showing the right-angled obsessed Churchmen don’t have the only workable glass bead game in town.

Theirs is not the only dogma or orthodoxy we might engage with.

In sum, Bucky spells out an alternative namespace wherein the term “dimension” is used differently than we’re taught in most schools. 4D stands for primitive volume, res mensa, without the energy involvement that gives res extensa (old Cartesian era terms). Then “frequency” adds the nuance of energy content, converting what’s 4D Platonic into something inherently spatiotemporal and/or experiential in some way i.e. involving energy (4D+).

So what if some dogmatists regard Synergetics as heresy? Their defensive “only room for one ruler” reflex has so far failed to stop a competing memeplex from spreading. Our freedom to explore this alternative namespace has not been curtailed.

In the newer more self-aware versions of STEM curricula, more mathematicians actually walk their talk and admit that axioms are the basis of the many metaphorical sandcastles along the beach (belief systems, churches): you can have any number of them, and yet no single one of them is the one and only Cathedral, insofar as our shared universe of discourse is concerned.

We have room for many maths (I favor the English plural over the singular here).

XYZ is under no threat whatsoever from Bucky’s Private Language, still spreading by osmosis through the Humanities, as a part of American literature. Its readership develops a healthy appreciation for sphere packing and polyhedrons, a mindset conducive to thinking about science, along with a flexibility many Stemites won’t have, given their mental stuckness in established dogma.

We still have a lot of eggs in the XYZ basket, like in Heidegger, and there’s no “winner takes all” plot to this soap opera. Positive synergy is what we seek. Let the glass bead game continue! We’re looking forward to what’s ahead.

Our ability to benefit from having alternatives requires a modicum of clarity to see that we have alternatives in the first place.

Letting Bucky take the stage a bit enhances the curriculum. Maybe he’s not the crackpot WIRED made him out to be? Maybe intellectual laziness is more to blame?

Nor is XYZ under any threat from Data Science, which is nowadays flooding the public namespaces with thousands upon thousands of dimensions, which simply mean “independent features of a specimen” such as weight, color, age etc.

In machine learning, every pixel in a picture may constitute its own dimension.

Part of the reigning orthodoxy was that especially gifted individuals are especially able to “see” in higher dimensions.

Spatial dimensions higher than three cannot normally be labeled “metaphorical” as in the philosophy of mathematics, the Platonic Forms are all actually present (versus “present” by analogy).

As mortal humans, the dogma goes, our sensory apparatus inhibits us in some way from detecting the multiple dimensions that are really there. We’re up against some kind of physical limitation in only being able to see a hypercube as a 3D projection. That “metaphor and analogy” is what transcends literally three dimensions sounds like too much of a concession, at least to those of a nominalist bent. Mathematics is not supposed to contain metaphors, as those go against “logical rigor” in some unspecified way.

Such was the influence of Edwin Abbott’s Flatland on everyday thinking, likewise mathematical satire.

Having satirized XYZ in his own thinking, accusing it of being “awkward”, Fuller took a different path, switching to “powering” over “raising the dimensionality of”. To each his or her own, right? Live and let live.

Then of course we have Fractional Dimensions as yet another namespace, which has gathered steam over the years, thanks to systems science. All of these namespaces are welcome and pose no threat to one another in pure principle.

However, some polemics may be in order just to keep them healthily distinguishable. We gain by keeping our namespaces from intermingling, willy-nilly.

There’s nothing gained by hitting the blender button and trying to pretend the Synergetics 4D is just an incoherent blend of Einstein’s 4D and Coxeter’s 4D.

A lot of Stemites I know never got very far with Synergetics because they lacked the necessary training in the Humanities.

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